Board 33

Board 33
Neither vulnerable

♠ K Q 5 ♥ A K Q 2 ♦ 6 2 ♣ K J 8 4

Partner passes in first seat, RHO opens one diamond, and I double. LHO passes, partner bids one spade, and RHO bids two diamonds. I double, showing a game invitation with only three trumps (and without five hearts). With a game invitation and four trumps, I would bid three spades. There was a time when two spades showed that hand. But today it's common to play that, if RHO bids, two spades simply confirms a fourth trump and doesn't show any extras.

Partner bids two spades over my double, and I pass. I have pretty much exactly what partner expects me to have. For the record, I would play a new suit by advancer in this auction, even a reverse, as simply scrambling for a better spot--not as a progressive move. For example, if advancer was 3-3-4-3 with zero high-card points, he would bid one heart over the first double. When I double again, he knows hearts are a three-three fit, so he should bid two spades, hoping to catch me with a four-card suit there.

West leads the ace of diamonds against two spades. Partner and I apparently don't see eye-to-eye about this auction:

	NORTH ♠ K Q 5 ♥ A K Q 2 ♦ 6 2 ♣ K J 8 4
	SOUTH ♠ A J 8 3 2 ♥ 10 7 ♦ 10 9 3 ♣ Q 9 2

West	North	East	South
			Pass
1 ♦	Double	Pass	1 ♠
2 ♦	Double	Pass	2 ♠
(All pass)

If the queen of clubs were the king, I assume partner would have bid two spades on the first round. So I don't quite understand why he wouldn't just bid four spades after my second double. I play the diamond deuce--seven--three. West continues with the king of diamonds--six--jack.

I know this is a doubleton, but does West?  Shouldn't East have echoed with jack doubleton? That depends on the context. Sometimes dropping an honor at trick one shows the next lower honor in case partner wants to underlead.  But West has no conceivable reason to underlead here.  And three rounds of diamonds, forcing dummy to ruff with an honor, looks like an attractive defense.  So I think East's carding should indicate whether or not to do that.  I believe, with this dummy, that seven-jack should deny a doubleton.

I drop the nine, hoping to make it appear that East had J107. But my opponents are on the same wavelength. West continues with the queen of diamonds. Of course. I now recall an earlier board where West played low, then jack with queen-jack third. I guess the way Jack cards, this has to be jack doubleton. I ruff with the king of spades, and East discards the five of clubs.

On the queen of spades, East plays the six, and West plays the four. On the next trump, East plays the nine, I play the jack, and West discards the four of diamonds. I don't really think I can coup East, but it's worth a try. I play the deuce of clubs; West plays the ace; East, the seven. West taps me with a diamond while East pitches the ten of clubs. I suspect East was 4-4-2-3 and that was his last club. If he was 4-3-2-4, he would have been pitching hearts.  But it doesn't hurt to try. I play a club to the jack, and East ruffs. Making three. We obviously belong in game. We are lucky we can't make it.

	NORTH ♠ K Q 5 ♥ A K Q 2 ♦ 6 2 ♣ K J 8 4
WEST ♠ 4 ♥ J 8 3 ♦ A K Q 8 5 4 ♣ A 6 3		EAST ♠ 10 9 7 6 ♥ 9 6 5 4 ♦ J 7 ♣ 10 7 5
	SOUTH ♠ A J 8 3 2 ♥ 10 7 ♦ 10 9 3 ♣ Q 9 2

It was wrong for East to split. I'm not looking at his hand, so I have no reason to finesse the eight of spades. As it happens, I can't coup him, but he doesn't know that. I can coup him if I have jack third of hearts and a doubleton club. I play a club.  If West wins and taps me, I can cash whatever four cards I need to to come down to the requisite ending.  If West ducks, I can cash four hearts and exit with a club, forcing West to coup his partner.

But East doesn't need to work all that out. Unless he's sure he can't be couped, he should just assume he can be and play low. Defenders sometimes seem to be afraid that declarer has x-ray vision. Computers in particular have this problem, because they're programmed to look for a way to beat the contract on a double-dummy basis.

At the other table, the auction begins the same way, but my hand raises two spades to three. At least Jack is in tune with himself. If South isn't going to bid any more than two spades with the hand he held, North has to bid again. But this can't be the right way to play. There's no reason to believe South has any high cards at all, and no reason to believe he has a fifth spade. Two spades could already be too high. It's true that you should never assume partner has a yarborough in a jammed auction. Over a pre-empt, for example, you should simply credit partner with about six high-card points and bid those values for him. But we have plenty of room here, so there is no reason for North to bid his partner's cards.

South, of course, carries on to four spades, which is down one. We pick up five imps, undeservedly raising our constructive bidding average. Actually, maybe it's not undeserved. Their final contract was better than ours, but their auction had twice as many bad bids.

Me: +140
Jack: -50

Score on board 33: +5 IMPs
Total: +81 IMPs

Boards 1-32

I've played 32 boards in this match so far, and it has been rather one-sided. I've won 99 imps and lost only 23. I'm going to take a break at this point to analyze the match and try to determine where the IMPs have come from. I could simply look at each board and try to determine the cause of each swing. But I don't trust myself to be objective. Instead, I'm going to try to devise a way for an impartial observer, one who perhaps doesn't even know anything about bridge, to analyze the results. To begin with, I'll establish four categories of bridge skills: declarer play, defensive play, constructive bidding, and defensive bidding. Then I'll rate my performance against Jack in each of these categories.

To rate myself in declarer play, I'll look at each board where Jack and I declared the same contract from the same seat and average the IMP results. Obviously, when looking at an individual result, declarer-play skill is not the only determinant. Luck plays a part as well. I might take a 70% line and go down while Jack takes a 50% line and makes it, or vice versa. Luck may also take the form of each table's having a different auction that gives different information either to declarer or to the defense. But, in the long run, these factors should even out. If we look at enough boards, the average score should be a fair measure of my edge over Jack whenever I declare a hand. (When I say 'edge over Jack,' that does not necessarily imply superiority. This edge could, after all, be negative. But I certainly hope it isn't.)

Next, I'll look at each board where Jack and I defended the same contract from the same seat. I'll take the average of those results as a measure of my edge as defender.

Bidding is trickier. I toyed with various ways of deciding whether bidding was constructive or defensive. Originally, I thought of making that distinction based on the auction itself, but I realized that was wrong. "Constructive bidding," after all, involves getting to the right contract when the hand belongs to your side, and "defensive bidding" involves interfering with your opponents' ability to do the same. So the distinction really depends not on whose hand you think it is during the auction but on whose hand it actually is. One objective measure of that is the balance power. Accordingly, I'll define constructive bidding as bidding on hands where we have 21 or more high-card points and defensive bidding as bidding on hands where we have 20 or fewer high-card points.

For constructive bidding, I start by taking the score on each deal where our side has 21 or more high-card points. I now normalize each score by deducting what I have determined is my edge in cardplay. If I declared, I deduct my edge in declarer play. If I defended, I deduct my edge in defense. I then take the average of these normalized scores. The result should be a fair measure of my edge in constructive bidding. I then do the same thing for hands with 20 or fewer high-card points to determine my edge in defensive bidding. If I haven't made any mistakes in my calculations, the average of my two cardplay edges plus the average of my two bidding edges should be roughly equal to my average IMP gain per board. Perhaps I should run this method by the Gargoyle quant department to make sure it makes sense. But they're busy developing trading algorithms for us, so I don't want to distract them.  I did run it by Jeff Rubens.  He was of the opinion that it would take an enormous number of boards for the extraneous factors to average out to zero.  He may be right, but if so we should be able to tell that by calculating the 95% confidence levels.

Before I calculate the averages, let me make some predictions. I suspect that my biggest edge over Jack will be in defensive card play. The way Jack plays is to generate random deals and to select the card that works most often across all these deals. The difficulty in this method is in choosing constraints for generating the deals, particularly constraints derived from other players' decisions or, what is often more important, from their failure to make certain decisions. "Why didn't declarer attack clubs at trick two?" "Why isn't declarer trying to ruff a heart in dummy?" "Why did declarer come to his hand to play trumps instead of playing them from the dummy?" These are the kinds of questions expert defenders are always asking themselves. Jack is, so far as I know, incapable of this kind of detective work. And defensive card play is where detective work is most important. Some of my edge disappears because I have Jack as a partner, and our ability to communicate is limited. (I don't mean just in signaling. I also mean communicating simply by the act of adopting one defense over another.) If I were defending with a human partner, I would expect our edge on defense would be enormous.

I suspect my next biggest edge over Jack will be in declarer play. Again, the ability to draw both positive and negative inferences is critical. But there are a number of declarer problems where there are few inferences to draw and where you must simply choose the highest percentage action among a variety of choices. Jack may well find those problems easier than I find them, particularly if the constraints are hard to quantify.

Next would be defensive bidding. Defensive bidding requires an understanding of how alternative actions on your part might make life easier or harder for your opponents. I don't think Jack has this understanding. Constructive bidding, on the other hand, I would expect Jack to be quite good at. Generating random deals for partner consistent with his auction and determining what your percentage action is over that set of deals is something Jack should be able to do faster and better than I do. Where I might still have an edge is in determining more accurately what "consistent with his auction" means. Jack is better at positive inferences than at negative ones. I also think Jack isn't good at factoring in how partner will react to an auction you are considering. It's one thing to conclude that you have an 80% chance at a game. It's another to conclude that you should still just invite because your chance opposite hands where partner would refuse your invitation is only, say, 30%. Jack seems just to bid game in those situations. At least that's been my impression.

Thirty-two boards isn't a large sample. But let's see where we stand anyway. Here are the averages, along with their 95% confidence levels:

Summary of Boards 1-32

Category	Average IMPs
Defensive play	+1.9 ±2.7
Declarer play	+0.8 ±1.6
Defensive bidding	+1.5 ±2.9
Constructive bidding	+0.9 ±2.7
Total	+2.4 ±2.0

As you can see from the large confidence levels, this really doesn't mean anything yet. I'll post these results again after 64 boards.

Board 32

Board 32
Opponents vulnerable

♠ 2 ♥ J 9 4 3 2 ♦ J 2 ♣ K Q J 10 5

Partner opens one diamond in second seat. I respond one heart, and partner bids two spades. Paradoxically, the easiest way to find a club fit in this auction is not to bid them.  If you bid three clubs, you could have a 3-5-2-3 pattern without a club stopper. So partner, with a 4-1-5-3 including the ace of clubs, will simply bid three notrump.  If you bid two notrump, however, partner should bid three clubs with that hand (unless his singleton heart is an honor). Then the fun will start.

I bid two notrump, partner raises to three, and I pass. West leads the seven of clubs:

	NORTH ♠ A K 7 6 ♥ 6 ♦ A K 9 5 4 ♣ A 6 3
	SOUTH ♠ 2 ♥ J 9 4 3 2 ♦ J 2 ♣ K Q J 10 5

West	North	East	South
Pass	1 ♦	Pass	1 ♥
Pass	2 ♠	Pass	2 NT
Pass	3 NT	(All pass)

You already know I disapprove of three notrump.  But I also have my doubts about two spades. Make one of the small diamonds a small heart, and you would rebid two notrump. This is not as good a hand (at least not after a one heart response). Why drive to game with this hand and only invite with the better hand?  I think many players tend to be too aggressive with their jump shifts as opener.  If responder can't find another bid over one spade, I wouldn't worry too much about missing a game.  And if he does find another bid, it's apt to tell you more about his hand than a forced rebid over two spades would.

Now for the play. Nine tricks are easy. Let's see what I can do about taking ten. I can't afford to duck a diamond, since the opponents might be able to take four heart tricks. I can, however, afford to duck a heart. I win the first trick with dummy's ace of clubs. East plays the four, and I play the ten. (Not that I expect to be using dummy's six as an entry later. But why burn bridges?) I play dummy's heart six-five-nine-ten. West continues with the king of hearts. I pitch a diamond from dummy. (The fourth spade isn't important, but the opponents don't know that. If someone has jack-fourth or ten-fourth, he may be afraid to pitch one for fear I have queen doubleton.) East plays the eight; I play the deuce.

West now shifts to the four of spades. What's going on in hearts?

____

It appears hearts were four-three, with West having king-queen-ten and East having the ace. But I can't tell who had the four-card suit. Either way the suit is blocked.  So I could win the spade ace, cross to my hand and play another heart, hoping to squeeze someone in hearts and diamonds. But there are several flaws to that plan. For one, the hand with the long heart is unlikely to have the sole diamond stopper.  For another, if the hand that wins the trick has the last club, he can destroy the squeeze by playing it.  This kills the entry to my hand before I can cash dummy's spade king. And finally, I only suspect hearts were four-three. I have no real assurance that they weren't five-two. I'm not willing to go down in pursuit of this overtrick.

I play the spade ace--three--deuce. I play a club to my hand. East plays the nine; West, the deuce. On the third club, West follows with the eight. (I note for future reference that he leads second highest from three small against notrump.) East pitches a dramatic ace of hearts. I suppose that means he was 4-3-4-2, probably with jack-fourth or ten-fourth of spades, so holding on to dummy's spades paid off. I cash my last two clubs, pitching dummy's spades. West pitches the seven of hearts and the eight of spades. East pitches the five and ten of spades. If I've read this hand correctly, his remaining spade should be the jack.

I can now safely float the jack of diamonds. If West has ten doubleton of diamonds, I'll make an overtrick. I lead the jack, and West covers with the queen. Oh, well. I play the ace and East drops the seven. An interesting card. I guess he's giving count. I cash dummy's spade, East plays the jack (I knew it!); West, the nine. I'm down to this position. The remaining diamonds are the ten, eight, six, and three.

	NORTH ♠ -- ♥ -- ♦ K 9 5 ♣ --
WEST ♠ Q ♥ Q ♦ x ♣ --		EAST ♠ -- ♥ -- ♦ x x x ♣ --
	SOUTH ♠ -- ♥ J 4 ♦ 2 ♣ --

If West has the singleton six or eight of diamonds remaining, I can lead the nine of diamonds from dummy, endplaying East for an overtrick and punishing him for wasting that seven of diamonds. Unfortunately, that's a pleasure I'll have to forgo, since if West's singleton diamond is the ten, I'll go down. I would lead the nine of diamonds at matchpoints.  Ostensibly, the odds in favor of the endplay are two to one.  In practice, I think they're considerably higher than that.  For West to have a singleton ten, East would have to have played the seven from eight-seven-six-three, and most players just don't do things like that.  In fact, I think it's so unlikely that I would consider the play at IMPs if I were losing the match.  The utility curve is not linear, and I suspect that my utile expectation is positive if I'm losing.  But under the current conditions,  I've already thought about it too long.  I cash the king. Making three.  As I suspected, the nine would have worked:

	NORTH ♠ A K 7 6 ♥ 6 ♦ A K 9 5 4 ♣ A 6 3
WEST ♠ Q 9 8 4 ♥ K Q 10 7 ♦ Q 6 ♣ 8 7 2		EAST ♠ J 10 5 3 ♥ A 8 5 ♦ 10 8 7 3 ♣ 9 4
	SOUTH ♠ 2 ♥ J 9 4 3 2 ♦ J 2 ♣ K Q J 10 5

At the other table, South jumps to three notrump over two spades, and North passes. This is wrong. Jumps to three notrump in forcing auctions should show extras. Even if you are generally a fan of fast arrival (and I'm not), fast arrival makes no sense when choosing between two notrump and three notrump. You frequently need two notrump as an exploratory move to allow partner a chance to finish describing his hand, so you can't afford to play that it promises extras. Three notrump, then, should promises extras. It should show a hand where you want to suggest slam but where you're unwilling to bid past game to do so.

Jack makes a different try for the overtrick. He wins the spade lead, cashes the diamond ace (trying to drop a stiff queen I suppose), then runs clubs. West pitches the seven and ten of hearts. East pitches the five of hearts and two low spades. With only three hearts outstanding, it is safe for declarer to lose a diamond. So when he leads the jack and West covers with the queen, he ducks. When the ten of diamonds doesn't drop, he, like me, is held to three.

Me: +400
Jack: +400

Score on Board 32: 0 IMPs
Total : +75 IMPs

Board 31

Board 31
Our side vulnerable

♠ J ♥ K 10 9 7 4 ♦ Q J 9 7 2 ♣ 9 4

No action appeals at this vulnerability, so I pass in first seat. LHO opens one notrump, and partner overcalls with two spades, Cappelletti, showing spades and a minor. I can guess which one.

I'm begrudgingly playing Cappelletti with Jack (sorry, Mike), because experience has shown me that it doesn't understand Astro. Why anyone would play any conventional method over the opponents' notrump openings other than Astro is beyond me. It's the only method that allows you to bid with 5-4's or 4-4-4-1's and has follow-ups to let you sort out the overcaller's relative suit lengths. Perhaps that's actually why more people don't play it. Other methods generally don't require you to define your bids past the overcaller's initial action. Astro requires understandings about the later auction and about how to scramble if the opponents start doubling. Most casual partnerships aren't up to this.

It's also difficult to find documentation on how Astro works. The best write-up can be found in "Bridge in the Sixties, Part III," by Rosler, Stern, and Allinger, in the December, 1960 issue of The Bridge World and in "Astro Revisited," by the same authors, in the January, 1962 issue. Copyrights prohibit me from reprinting those articles. But, for anyone who is interested, I can offer you the pages from my partnership notebook: Partnership Notes on Astro.

Over two spades, RHO bids three notrump, which denies either four hearts or a spade stopper. Three notrump ends the auction, and partner leads the six of spades:

	NORTH ♠ 10 4 2 ♥ Q 8 3 ♦ K 6 5 4 3 ♣ A 5
		EAST ♠ J ♥ K 10 9 7 4 ♦ Q J 9 7 2 ♣ 9 4

West	North	East	South
		Pass	1 NT
2 ♠1	3 NT2	(All pass)
1Spades and a minor
2No four-card heart suit, no stopper

Partner has from seven to nine high-card point. By the rule of eleven, I know declarer has three spades higher than the seven. Declarer's likeliest pattern is 4-3-2-4. As I've mentioned in earlier posts, it often helps you to defend if you make a guess as to how the early play will go. If the play doesn't go that way, a bell goes off and you are alerted to the fact that there's something funny going on. If you don't make this guess, the bell may fail to ring and you may miss important inferences.

My expectation on this deal is that declarer will win the spade, cross to the club ace, and lead a heart to the jack. He will then cash whatever black-suit tricks he can, then play ace, king, and a diamond, forcing me to lead away from the king of hearts. If he doesn't have the heart jack, the play will probably go similarly except for the heart finesse.

Declarer plays the deuce from dummy, I play my jack, and declarer wins with the queen. Declarer leads the eight of diamonds to the king, partner playing the ten. That bell you hear ringing is what I was talking about earlier. He needs the diamond entry later for the endplay. Why is he playing a diamond to the king instead of a club to the ace? Perhaps he thinks partner's second suit is diamonds, and clubs is the suit he plans on endplaying me with? Even so, it doesn't hurt to use the club entry now. There must be some reason he can't afford to release the club ace, and it probably pertains to how he intends to play clubs later on. Until I know more about the hand, I won't know the reason, but I store this inference away for later consideration. For now, I drop the queen of diamonds to help partner count declarer's high cards. I don't want partner placing me with a club honor.

Declarer plays the four of spades from dummy. More tintinnabulation. If declarer wanted to play spades, why did he need to get to dummy to do so? Perhaps he wasn't using the diamond as an entry. Perhaps he was simply extracting an exit card from partner's hand, preparing to endplay him in hearts and clubs (thinking he has the heart king). But why cash the diamond king instead of the ace? Perhaps he needs the entry to his hand later? That might also explain why he didn't cash both diamonds.

One thing I can be fairly sure about: Either (1) declarer doesn't have the jack of hearts or (2) he has the jack but two heart tricks are sufficient to see him home. If he has the jack and needs three heart tricks, he wouldn't be playing this way. In case (2), we're not beating this, so I might as well assume case (1). I want partner to know that it's safe for him to lead hearts from his jack, so I discard the heart ten. Declarer plays the eight of spades (leaving him with one card higher than the seven), and partner wins with the king of spades. Partner cashes the spade ace. That makes eight high-card points (counting the jack of hearts). He has at most the jack of clubs in addition. I discard the diamond jack (present count), and declarer follows with the nine.

So partner began with six spades, which would make him 6-1-1-5. (Surely he wouldn't bother showing a second suit of jack fourth when he had ace-king sixth of spades.) Whatever declarer was up to, whether a fratricide squeeze or a throw-in, he was apparently hoping partner had started with only five spades. Declarer is going down at least one. How many tricks does he have after partner cashes his spades? One spade, two diamonds, one heart, and three clubs. That's only seven tricks, so we can conceivable beat this two. Declarer would have been able to squeeze me in the red suits, using the six of hearts in his hand as a threat, if he had saved dummy's diamond entry. The only way he can take an eighth trick now is in the club suit. If he has KQ108, he can either cash his tricks, playing me for jack doubleton or he can lead the ten from his hand, playing me for nine doubleton. There's not much I can do about that except never pitch a club. Technically, I can afford one pitch from either holding. But, psychologically, declarer is probably more apt to play me for the jack than the nine if I never pitch one. It turns out it doesn't matter. Declarer doesn't have the ten of clubs, and, eventually, he finishes down two. The full deal:

	NORTH ♠ 10 4 2 ♥ Q 8 3 ♦ K 6 5 4 3 ♣ A 5
WEST ♠ A K 7 6 5 3 ♥ J ♦ 10 ♣ J 10 8 7 6		EAST ♠ J ♥ K 10 9 7 4 ♦ Q J 9 7 2 ♣ 9 4
	SOUTH ♠ Q 9 8 ♥ A 6 5 2 ♦ A 8 ♣ K Q 3 2

I'm still not sure what declarer was up to. At the other table, declarer cashes both diamonds. (That East played the deuce under the king. Perhaps my declarer was planning on cashing both diamonds, but, for some reason, he changed his mind when he saw my queen.) West pitches a club. Declarer now cashes three clubs and exits with a club, hoping to endplay West in hearts for down one. It doesn't work. Down two for a push.

A good blog deal, but a lousy problem hand. I had no critical decisions at all, but it was still a fertile deal for discussing what to think about as you defend.

Me: +100
Jack +100

Score on Board 31: 0 IMPs
Total: +75 IMPs

Board 30

Settle in. This one gets involved.

Board 30
Neither vulnerable

♠ A 9 7 3 2 ♥ 7 3 ♦ A Q 9 ♣ Q 10 5

RHO passes, and I open one spade. LHO bids three diamonds, and partner bids four spades. Everyone passes, and LHO leads the six of hearts:

	NORTH ♠ Q 10 8 ♥ A J 10 5 ♦ 6 3 ♣ A 8 7 6
	SOUTH ♠ A 9 7 3 2 ♥ 7 3 ♦ A Q 9 ♣ Q 10 5

West	North	East	South
		Pass	1 ♠
3 ♦	4 ♠	(All pass)

It's going to be hard to make this with at least one loser in every suit. I don't care much for the four spade bid. I would have made a negative double, giving partner a chance to pass with four diamonds or with a "5332" with three diamonds (like the hand I held). If partner passes, each side has at most an eight-card fit, so the Law of Total Tricks suggests that there will be only 16 tricks available. In other words, if we can make four spades, they're down three in three diamonds. The big gain comes when we can't make four spades. Then three diamonds rates to be down two. Even if the Law is off by a trick, on balance we're better off defending when both sides have an eight-card fit.

But my task now is to make four spades. To do that, I'm going to have to start by bringing home the spades, which means I need to find West with a singleton jack. If I survive that hurdle, I need to find a way to dispose of my second diamond loser. One possibility is to play West for honor third of hearts. I can finesse at trick one, losing to East's honor. (This should be safe. The lead is unlikely to be a singleton, since East didn't open two hearts.) Later, I finesse again and pitch a diamond on the heart ace. That's a possibility, but a remote one. West has no particular reason to have made such an aggressive lead.

Another possibility is to play for three club tricks. Unfortunately, I'm going to have to start clubs from my hand. I need dummy's heart entry to take the trump finesse, and East can cover the last trump if he doesn't want me in dummy. I do have a variety of ways to take three club tricks, all involving playing East for a doubleton that includes the nine. I could, for example, play East for king-nine doubleton: float the queen, then lead the ten to pin his nine. Leading the queen also works if he has jack-nine doubleton or nine doubleton. If West covers my queen, I take the ace, then lead a club to my ten. If West has jack-nine, dummy's clubs are good. If he has nine doubleton, West can duck, but a third club endplays him. He will have to lead another club for me or lead into my ace-queen of diamonds. The important thing to notice is that this line requires West to be 1-2-6-4. I don't have to commit myself yet. I can wait and see what heart spot West plays when I lead a heart toward dummy. If he plays up, I know he can't be 1-2-6-4, so I might as well fall back on the heart finesse.

I play the jack of hearts from dummy. East wins with the queen, and I drop the seven. East shifts to the five of diamonds. If I knew I was going to take the heart finesse, I could afford to hop with ace in case West has seven diamonds. But to leave open the possibility of developing clubs, which requires West to be 1-2-6-4, I must duck this. And I must play the nine to retain my tenace for the endplay I envisioned above. West wins with the ten and returns the deuce of diamonds, which East ruffs with the five of spades.

So much for the endplay. West can't be 1-2-6-4 anymore, so I'm back to relying on the heart finesse. If West does have the heart king and a singleton jack of spades, I still have a shot to make this as long as East doesn't play a heart now. Say he plays a spade. I ride this around to West's jack and dummy's queen. I cash the club ace, then run spades and cash the diamond ace. East, with his four small hearts and club king, is squeezed. It may seem like a vain hope that East won't play a heart from four small to break up the squeeze. But it's not necessarily the right play. From his point of view, I could have,

♠ A J 9 x x x ♥ x x ♦ A Q x ♣ Q x.

If so, I have a guess between finessing the heart or playing for a heart-club squeeze against East. Returning a heart takes my guess away.

East shifts to the six of spades, keeping my hopes alive. I play the deuce. Unfortunately, West wins this trick with the king of spades, so I'm down. My job now is to hold the undertricks to as few as possible. I unblock the ten from dummy. West plays the king of diamonds, which I ruff with the queen, as East pitches the deuce of clubs.

If East is pitching a club, he probably doesn't have five hearts. That means West started with three, giving him either 1-3-7-2 or 2-3-7-1. I need to decide which so I know whether to finesse East for the jack of spades or not. The first pattern is more likely a priori, since the defenders have fewer spades than clubs. In addition, West might have led a singleton club (although it's not clear how valid an inference that is when he has king doubleton of trumps). In any event, I'm going to play West for 1-3-7-2 and finesse the spade. The next problem is how to hold myself to one club loser. West is unlikely to have a second side king, so I could just play ace and a club toward my queen. But West did pre-empt opposite a passed hand, so his pre-empt might be little flaky. It would be nice if I could find a way to make it even if West does have a doubleton king of clubs. How about a trump squeeze? I could cash all my trumps but one to reach this position:

	NORTH ♠ -- ♥ A 10 5 ♦ -- ♣ A 8
	SOUTH ♠ 2 ♥ 3 ♦ -- ♣ Q 10 5

If East is down to a doubleton heart, I can establish a heart trick by ruffing. If he is down to a doubleton club, I can play ace and a club. I don't care who has either king.

I play a trump, intending to finesse. East plays the jack, I play the ace, and West follows. So West did start with 2-3-7-1? Maybe it's a stiff king of clubs and I can get out for down one. I cash three more spades. East pitches three clubs, so I play ace and a club. Down two. It turns out West was void in clubs:

	NORTH ♠ Q 10 8 ♥ A J 10 5 ♦ 6 3 ♣ A 8 7 6
WEST ♠ K 4 ♥ 9 8 6 4 ♦ K J 10 8 7 4 2 ♣ --		EAST ♠ J 6 5 ♥ K Q 2 ♦ 5 ♣ K J 9 4 3 2
	SOUTH ♠ A 9 7 3 2 ♥ 7 3 ♦ A Q 9 ♣ Q 10 5

Against a human West, I might have suspected a club void when he led the deuce of diamonds. But I haven't seen any evidence that Jack even knows what a suit-preference signal is.

At the other table, the auction and lead are the same. Declarer loses the first trick to the queen of hearts, but hops with the ace when East shifts to a diamond. I'm not sure what his plan is, because it doesn't get very far. At trick three, he leads a club from his hand, and West ruffs. West now plays the king of diamonds. Surely this is wrong. He wants his partner to ruff this and play another club, so why not lead low? East ruffs anyway and plays another club. West ruffs with the king of spades and plays a third diamond. Declarer guesses correctly to ruff high and finesse East for the spade jack. Down one.

I'm not being facetious when I say he guesses correctly. If West did have king-jack of spades, he should still ruff with the king and play a third diamond. Ruffing with the jack dooms his king of spades once East can't overruff dummy. Not leading the third round of diamonds at all doesn't help, since there is no benign explanation for failing to lead a diamond. The only way to score two trump tricks is to falsecard.

I was curious whether Jack would find this mandatory falsecard. I edited the deal, switching West's four of hearts for the spade jack, and replayed it. The bidding and play were the same up to the point where East led the second club. Not only did West not ruff with the king, he didn't ruff at all!

I backed up the play and forced West to ruff (incorrectly) with the jack, then lead a diamond. I wanted to see if South would drop the king when East failed to overruff the dummy. He didn't. He took the finesse into West's stiff king. That surprised me. I can understand the mandatory falsecard's being beyond Jack's capabilities. But I should think that once East fails to overruff the diamond, Jack would exclude layouts from its universe where East holds the spade king. Computer programs play much better today than I would have thought possible ten years ago. But they still have a long way to go. Although, who am I to talk? I went down a trick more than Jack.

One last point. For those of you who were skeptical, may I point out that North would have done six IMPs better had he made a negative double?

Me: -100
Jack: -50

Score on Board 30: -2 IMPs
Total score: +75 IMPs

Board 29

Board 29
Both sides vulnerable

♠ 4 3 ♥ J 9 7 6 4 ♦ Q J 7 6 ♣ 5 4

Partner passes, and RHO opens one club. I pass, LHO passes, and partner balances with a double. RHO bids one spade. If partner weren't a passed hand, I would bid two hearts. But, since the opponents have at least 25 high-card points between them, I'd just as soon not prod them. If I bid two hearts, LHO will probably compete with two spades or three clubs, and RHO will go on to game. If I pass, perhaps LHO will pass also or give a preference to two clubs, which doesn't promise any values.

Wishful thinking. I pass, and LHO bids three spades. RHO, of course, bids four.

I suppose partner is either 3-4-4-2 or 3-4-5-1. (He might have balanced with one heart with 3-5-4-1.) Tapping declarer seems like our best chance to beat this, and diamonds is a more likely tap suit than hearts. It's hard to imagine a three-spade bid that includes an uncertain value like the king of diamonds, so I see no point in leading the queen. If dummy has, say, ten-fourth of diamonds, it may work out better to hold on to my honors. When you go for a tap, it's often a good idea to lead as if you were leading against notrump. (After all, tapping out declarer amounts to converting the contract to notrump.) This means leading low from two-honor sequences as well as underleading aces to preserve communication. Accordingly, I lead a third-best seven of diamonds.

	NORTH ♠ 8 7 5 2 ♥ A 8 ♦ 8 4 3 2 ♣ J 10 2
WEST ♠ 4 3 ♥ J 9 7 6 4 ♦ Q J 7 6 ♣ 5 4

West	North	East	South
		Pass	1 ♣
Pass	Pass	Double	1 ♠
Pass	3 ♠	Pass	4 ♠
(All pass)

Declarer plays the deuce from dummy; partner plays the king; declarer, the ace. 3-4-5-1 is out. Partner is apparently 3-4-4-2. Declarer plays the club king. I suspect declarer is more interested in the club count than partner, so I play the four. Declarer plays the deuce from dummy, and partner plays the seven. Declarer  plays the deuce of hearts--four--ace--five, then leads dummy's jack of clubs. Partner plays the queen; declarer, the ace.

Partner has shown up with five high-card points, so he has at most six left, at least two of which are in hearts. It's going to be hard to find four tricks. Declarer leads the nine of spades--five--deuce--queen. This doesn't add up. Declarer wouldn't be playing trumps this way unless he were missing another spade honor. So partner must have the king of spades as well, giving him a 12-count:

♠ K Q x ♥ Q 10 x x ♦ K 10 9 x ♣ Q x

It's a rather soft 12-count, so there's nothing wrong with choosing not to open.  Still, it might have made it a little harder for the opponents to reach four spades if he did.

Partner plays the ten of diamonds, which declarer ruffs with the ten of spades. I'm defending double-dummy at this point, and we are down to the following position:

	NORTH ♠ 8 7 5 ♥ 8 ♦ 8 4 ♣ J
WEST ♠ 4 ♥ J 9 7 6 ♦ Q J ♣ --		EAST ♠ K 6 ♥ Q 10 3 ♦ 9 5 ♣ --
	SOUTH ♠ A J ♥ K ♦ -- ♣ 9 8 6 3

Declarer could cash the ace of spades and claim five, but he needs to worry about 4-1 trumps. He plays a club, which I ruff. I tap him again with the queen of diamonds. He cashes the trump ace, then plays another club, pitching dummy's last diamond as partner ruffs. Declarer now has the rest. Making four.

	NORTH ♠ 8 7 5 2 ♥ A 8 ♦ 8 4 3 2 ♣ J 10 2
WEST ♠ 4 3 ♥ J 9 7 6 4 ♦ Q J 7 6 ♣ 5 4		EAST ♠ K Q 6 ♥ Q 10 5 3 ♦ K 10 9 5 ♣ Q 7
	SOUTH ♠ A J 10 9 ♥ K 2 ♦ A ♣ A K 9 8 6 3

At the other table, the auction begins the same way, but my hand bids two hearts over one spade. I've already explained why I think that's wrong. North raises to two spades (why he bids only two over two hearts but three over a pass I can't say), East bids three hearts, and South bids four spades.

West leads the four of hearts. This doesn't give declarer quite as much trouble as a diamond lead. Declarer wins in his hand, then cashes the club king, unblocking dummy's ten. He cashes the spade ace, then the club ace, unblocking the jack. Once clubs split, he switches back to spades and makes five. I suppose this is the best line. Assuming, from the double, that East has at least three spades and at most two clubs, it works anytime clubs come home, even against 4-1 trumps, and it works any time trumps are 3-2. It's hard to see how to improve on that. While it seems natural to cross to dummy and play the second club toward your hand, it doesn't actually accomplish anything. If East has a singleton club and four trumps, you're still going down. [It get's complicated, but it turns out this isn't true.  See poohbear's comment below.]

Note, by the way, that I was right yesterday.  Jack sees nothing wrong with passing his partner's opening bid with an ace.  Although, as Willenken would be happy to point out, he almost missed a game by doing so.

Me: -620
Jack: -650

Score on Board 29: +1 IMP
Total: +77 IMPs

Board 28

Board 28
Our side vulnerable

♠ A 9 4 ♥ Q 9 7 5 4 2 ♦ -- ♣ A 7 5 3

LHO opens one diamond, which is passed around to me. I bid one heart, and partner bids three notrump. With most weak notrumps, I would expect partner to bid merely two notrump. It's possible he has a hand at the very top of this range. More likely, though, he has a hand in the strong notrump range that was somehow unsuitable for an immediate one notrump overcall. And the likeliest reason for it to be unsuitable would be a singleton heart. This is a long-winded way of saying that correcting three notrump to four hearts might not be such a good idea. Still, passing three notrump with a diamond void doesn't feel right either. I decide to hedge my bets by bidding four clubs. If partner has the good weak notrump, he can bid four hearts on his doubleton. If he has a singleton heart and club support, he can bid five clubs. And if he is stuck with, say, a 4-1-5-3 pattern, he can bid four diamonds, over which I will bid four hearts.

At least that's what I thought at the time. In retrospect, I don't think I gave enough thought to how five clubs would play. It's pretty hard to construct a 3-1-5-4 where five clubs is much of a contract. Nor am I sure how happy partner would be to let me play four hearts when he's 4-1-5-3. He might decide that, if I couldn't rebid hearts right away, I'm not willing to play opposite a singleton. Five clubs in a four-three fit rates to be a pretty ludicrous spot. If I could bid this hand again, I would just bid four hearts. I think I succumbed to flexing disease. But I needn't have worried. Partner bids four hearts over four clubs, and West leads the seven of spades:

	NORTH ♠ Q J 8 5 ♥ K 8 ♦ A J 10 4 ♣ Q J 4
	SOUTH ♠ A 9 4 ♥ Q 9 7 5 4 2 ♦ -- ♣ A 7 5 3

West	North	East	South
1 ♦	Pass	Pass	1 ♥
Pass	3 NT	Pass	4 ♣
Pass	4 ♥	(All pass)

I play the queen of spades; East plays the three. I still don't know how the spades lie. This carding is consistent with king-ten fourth in either hand or with West's holding a singleton. The only thing I can be fairly sure of is that spades aren't three-three. My first instinct is to ruff a diamond to my hand and play a heart to the king. But if this loses to the ace, I'm down. East will play a spade back, and I can't get to dummy to pitch my spade on the ace of diamonds. Some players will not pass partner's opening bid with an ace. (If you want to see Chris Willenken go berserk, just try passing his opening bid with ace.) But I know from past experience that Jack is not a member of this cult.

Do I really have to cash the ace of diamonds and pitch a spade now? On a strategic level, this seems wrong for a lot of reasons. It destroys my tenaces in two suits. It gives the opponents a suit they can tap me with. And it commits me to pitching on the diamond ace before I'm even sure what I want to pitch. If East has the king of clubs, perhaps I should be leading ace and another spade, eventually pitching two clubs on the jack of spades and the ace of diamonds.

I shouldn't have too much trouble making this hand if West has the king of clubs. In that case, it won't hurt to cash the diamond ace now and give up all these intangible strategic assets.  But if East has the club king and clubs aren't 3-3, I have two clubs losers. I would prefer, in that case, to maintain as much flexibility as possible. Perhaps I shouldn't worry too much about East's holding the club king. If he does, that means West has king fourth of spades, which increases the odds that his heart ace is doubleton. If I have only one trump loser, I can afford a second club loser. Instinctively, it still feels like the wrong thing to do. But analysis tells me it's right. So I cash the diamond ace--three--spade nine--diamond nine.

I lead the jack of diamonds (if East covers, this will serve to kill diamonds as a tap suit)--seven--heart four--diamond deuce. I now play the deuce of hearts--ace--eight--three. There could be a problem here. If this is a stiff ace, I have to hold myself to one club loser after all.

West continues the tap. He plays the five of diamonds--four--queen--five of hearts. It looks as if diamonds were 6-3, but I still don't know what's going on in the spade suit. In any event, I need to work on clubs. I play the three of clubs--king--four--six. West plays diamond king as East pitches the eight of clubs. I ruff, and I'm down to this position:

	NORTH ♠ J 8 5 ♥ K ♦ -- ♣ Q J
	SOUTH ♠ A ♥ Q 9 ♦ -- ♣ A 7 5

I play a heart to dummy's king. If they don't break, I intend to try to cash a club. If it cashes, I'm home. I can play dummy's last club, then a spade to my ace. East can ruff whichever of these he pleases, but, with no more diamonds, he can't tap me again. If he ruffs the first club, however, I'm going down, since the clubs are blocked. As it happens, everyone follows to the heart. My contract is now safe. I play the queen of clubs. Everyone follows to that as well. I play a spade to my ace, draw the last trump, and claim. Making five.

	NORTH ♠ Q J 8 5 ♥ K 8 ♦ A J 10 4 ♣ Q J 4
WEST ♠ 7 6 ♥ A J ♦ K 9 8 6 5 2 ♣ K 10 2		EAST ♠ K 10 3 2 ♥ 10 6 3 ♦ Q 7 3 ♣ 9 8 6
	SOUTH ♠ A 9 4 ♥ Q 9 7 5 4 2 ♦ -- ♣ A 7 5 3

Given that the king of clubs was on my left all along, I came a lot closer to going down in this contract than I would have liked. It occurs to me that I flexed in the auction but not in the play.  That's atypical of me on both counts.  I'm curious to see how Jack declares.

At the other table, the auction begins the same way, but South simply removes three notrump to four hearts. West leads the seven of spades, and dummy's queen holds. Jack plays a low diamond from dummy at trick two. East hops with the queen. Declarer ruffs and plays a heart. West rises with the ace and plays the six of spades--jack--king--ace. Declarer leads a heart to dummy to pitch a spade on the diamond ace, ruffs a spade to his hand, and draws the last trump. He's now down to this position:

	NORTH ♠ -- ♥ -- ♦ J 10 ♣ Q J 4
	SOUTH ♠ -- ♥ 9 ♦ -- ♣ A 7 5 3

He plays a club to dummy's jack, then plays the diamond ten, pitching a club from his hand. West is in with the king of diamonds but must either lead away from his king of clubs or play a diamond to dummy's jack, allowing declarer to pitch his club loser. Making the same five I made on a radically different line.

My opponent was certainly better placed than I was for the very reasons I was uncomfortable with my trick two decision: Diamonds weren't available to the defense as a tap suit, and retaining the spade tenace gave him some extra chances. Note that at the point he ruffed a spade to his hand, he had eight-five of spades in dummy behind East's ten-deuce. If he needed to, he could have established a spade trick by taking a ruffing finesse against the ten. This threat might have come into play if East hadn't hopped with the diamond queen at trick two.

All this sounds more like analyzing a chess position than discussing a bridge hand. Bridge analyses tend to deal with specifics. But sometimes, when there are lots of things that could go wrong and you don't know enough about the hand to know what to cater to, declarer play can become more chess-like than it normally is.

Is Jack's line better than mine?  Possibly.  This is, after all, exactly the kind of problem Jack should be good at solving: lots of possible layouts with constraints that are hard to quantify.  But the fact remains that if East had held the heart ace, Jack wouldn't have been very happy with his line.  So I'm still not sure.

Me: +650
Jack: +650

Score on Board 28: 0 IMPs
Total: +76 IMPs

Board 27

Board 27
Neither vulnerable

♠ K 10 9 6 4 ♥ A K 10 8 6 4 ♦ -- ♣ A Q

I open one heart in first seat. LHO and partner pass, and RHO balances with two diamonds. I could bid two spades, but three spades is more descriptive. Partner will know I'm 5-6 and have a rough idea of the strength of my hand. I can then leave further bidding up to him. Since our defensive prospects depend largely on his diamond holding, he will know better than I how high to bid.

LHO bids four diamonds, partner bids four hearts, and RHO bids five diamonds. This is precisely why I clued partner in. I pass the decision around to him, and he bids five hearts, which ends the auction:

	NORTH ♠ 8 3 ♥ Q J 9 7 5 3 ♦ 8 7 6 ♣ 10 8
	SOUTH ♠ K 10 9 6 4 ♥ A K 10 8 6 4 ♦ -- ♣ A Q

West	North	East	South
			1 ♥
Pass	Pass	2 ♦	3 ♠
4 ♦	4 ♥	5 ♦	Pass
Pass	5 ♥	(All pass)

I agree with partner that it's wrong to bid four hearts on the first round. If the auction were to continue, say, double--pass--four spades, he would have a problem. It would be a violation of captaincy to bid five hearts in front of me. For all he knows, his four heart bid gave the opponents a problem and they're stretching. On the other hand, he knows--but I don't--that five hearts is a cheap save. If he passes, I might sell out to four spades not because I think we can beat it but because I think five hearts would be too expensive. In general, you should pre-empt when you know how high you're willing to bid and you want the opponents to make the last guess. When your side is the one making the last guess, you want to keep the bidding low to get as much information as possible. This is apparently what partner had in mind when he passed one heart.

My preference, though, is to start with a simple raise. At least this lets partner know I have support, so my hand won't be a complete surprise if the auction gets high quickly. But I hope it doesn't. I hope the auction is slow and informative, so that I will be able to judge what the opponents can make. If I get really lucky, I may even be able to alert partner to what I have by taking an impossible auction. For example,

West	North	East	South
			1 ♥
Pass	2 ♥	Double	Pass
3 ♥	4 ♥

If you're generally disciplined about not bidding again unilaterally with a limited hand, this auction suggests something unusual. I believe it should suggest this hand, that is, a hand where you would consider bidding five hearts on the first round if it were defined as pre-emptive. Now, if the auction proceeds pass---pass--four spades, you have no need to bid in front of partner. You can pass and let him make the decision.

In any event, I find myself in five hearts. West leads the six of clubs, which makes it easy. I win with the club queen, play a heart to dummy, and lead a spade to my king. The ace is onside, so I make six:

	NORTH ♠ 8 3 ♥ Q J 9 7 5 3 ♦ 8 7 6 ♣ 10 8
WEST ♠ Q J 5 ♥ 2 ♦ A Q 4 3 2 ♣ K 7 6 5		EAST ♠ A 7 2 ♥ -- ♦ K J 10 9 5 ♣ J 9 4 3 2
	SOUTH ♠ K 10 9 6 4 ♥ A K 10 8 6 4 ♦ -- ♣ A Q

At the other table, the first round of bidding is the same, but South bids three diamonds at his next turn. I don't care for this. What's the point of telling partner you have a good hand without offering any idea of what it looks like? West bids four diamonds; North, four hearts; and East, five diamonds. South now has to make the decision himself, and he chooses to bid five hearts. Everyone passes.

At my table, West apparently wasn't worried about my clubs being as good as they were, so he attacked the suit at trick one. Against the less descriptive auction at the other table, West had more reason to shy away from leading clubs himself. He chose a low diamond, hoping to get his partner in for a club switch. This was a nice lead. Just change the layout slightly and it becomes the only lead to beat five hearts. Funny how the more descriptive auction elicits the more favorable lead. It doesn't usually work that way.

After the diamond lead, declarer ruffed, played three rounds of trumps ending in dummy, and led a spade to the king. Is this safe? Suppose West taps him with a diamond. He's down to one trump. He plays a spade. The defense wins and taps him again. If he can't establish the spades with one ruff, he needs to fall back on the club finesse.

There must be a sure-trick line. How about stripping the hand and playing a spade to the nine? Ruff the diamond, trump to dummy, ruff another diamond, trump to dummy, ruff the last diamond, trump to dummy. No, I didn't have a finger left for that last trump. That means I've played six trumps, and I'm now out of trumps in my hand.

How about this: Ruff the diamond, trump to dummy, spade to the nine (just in case West has a stiff ace). West wins and taps me. I play another trump to dummy. I now have two trumps left in my hand. I lead the spade eight. If East shows out, I play low and eventually ruff out West's remaining honor. If East follows low, I play the king. If West wins this, spades are no worse than four-two, and I have three entries left to my hand to establish a spade and get back to cash it. That works.

This is a flaw in Jack's technique for playing hands.  Remember that he deals out random hands and chooses the line that works most often.  If a line works on 100% of the hands he dealt, he'll choose it and may miss a better line that could be found by analysis rather than by trial and error.  While choosing a 99% line instead of a 100% line offends us purists, in practice the cost is small.  This technique should at least prevent him from taking a 70% line when an 80% line is available, something humans are quite capable of, particularly if the constraints are such that the odds are hard to calculate.

Me: +480
Jack: +480

Score on Board 27: 0 IMPs
Total: +76 IMPs

Board 26

Board 26
Both sides vulnerable

♠ A K Q J 9 ♥ Q 9 ♦ K 10 ♣ J 10 6 4

Partner opens one diamond in second seat. I respond one spade, and partner rebids one notrump. My instinct tells me this isn't quite worth a slam try, but let's try applying Culbertson's rule¹ just to make sure. To make slam virtually laydown, partner needs something like ace of hearts, ace-queen of diamonds, and king-queen of clubs.  Not only is that not a minimum, it's not even within his range. Suppose I give him a source of tricks, say ace-queen-jack fifth of diamonds.  He still needs the club ace and the heart king.  Again, not a minimum.  So I don't have an invitation.

The next decision is whether to look for three-card spade support or simply to bid three notrump. Extra high cards, running spades, high-cards in my short suits, and soft values in my side suit all point toward notrump. So I bid three notrump, which ends the auction.

	NORTH ♠ A K Q J 9 ♥ Q 9 ♦ K 10 ♣ J 10 6 4
	SOUTH ♠ 7 ♥ J 10 8 2 ♦ A Q 9 8 2 ♣ A K 5

West	North	East	South
		Pass	1 ♦
Pass	1 ♠	Pass	1 NT
Pass	3 NT	(All pass)

West leads the five of hearts. What card do you play from dummy?

----

Playing low would mark you with the jack, so you should play the queen. East wins with the ace and returns the six of spades--seven--three--nine. I wasn't expecting that. He must have a singleton ace of hearts. But what's the strategic goal of shifting to spades? If dummy had no entries, it might force me to cash spades before I was ready, either bollixing up a squeeze or simply forcing me to discard before I learn anything about the hand. But I suspect East led a spade just because it looked like the safest exit. Obviously he wouldn't want to play a diamond. But why not a club? A club might even be productive. I think it's a fair assumption East has the club queen.

My goal at this point is to take the rest of the tricks. One way to do that would be to drop the jack of diamonds, a 52% chance. But I think the odds the club queen is onside is much higher. True, the club finesse brings me only to eleven tricks. But the twelfth is always there provided I read the position correctly. I lead the jack of clubs--seven-five-nine. It appears East has Q732 and West has 98. I play another club--deuce--king-eight. I cash the club ace, and West pitches the three of hearts.

This is the wrong heart, by the way. After a fourth-best lead, you should play your lowest card next if you began with five, and you should play a card higher than the one you led if you began with six. (This is assuming you want to tell partner how many cards you began with, which isn't necessarily true of course.)   If you give present count, you create an ambiguity between four and five, since you would play up with either holding.  If you always play low, you create an ambiguity between five and six, since you would play down with either holding. In the suggested method, the ambiguity is between four and six, which is an ambiguity partner is likely to be able to resolve.

Anyway, I now have a classic double squeeze. East guards clubs, and West guards hearts. So no one can guard diamonds. I play a diamond to the king, run the spades, and claim. Making six.

	NORTH ♠ A K Q J 9 ♥ Q 9 ♦ K 10 ♣ J 10 6 4
WEST ♠ 8 4 3 ♥ K 7 6 5 4 3 ♦ J 4 ♣ 9 8		EAST ♠ 10 6 5 2 ♥ A ♦ 7 6 5 3 ♣ Q 7 3 2
	SOUTH ♠ 7 ♥ J 10 8 2 ♦ A Q 9 8 2 ♣ A K 5

As the cards lie, it doesn't matter.  But on a different layout, East might hold me to five by covering the jack of clubs with queen fourth.  After running the spades, I will be down to:

	NORTH ♠ -- ♥ 9 ♦ 10 ♣ 10 6
	SOUTH ♠ -- ♥ -- ♦ A Q 9 ♣ 5

If East originally guarded both minors, he will already have been squeezed.  But what if West guards one? If he guards diamonds, I can squeeze him by cashing the ten of clubs.  If he guards clubs, I have to cash diamonds to squeeze him.  I need to guess, and, since covering the jack of clubs with queen fourth is an unusual play, I would probably guess wrong.

At the other table, North rebids two clubs over one notrump, checking back for a spade fit. South bids two diamonds, and North bids three notrump. The first two tricks are the same, but declarer doesn't draw the same conclusion about the club queen.  He just cashes his tricks, banking everything on diamonds coming home, which they do. Making six for a push.

¹Culbertson's rule: Invite a game or slam only if partner's perfect minimum makes the contract virtually laydown.

Me: +690
Jack: +690

Score on Board 26: 0 IMPs
Total: +76 IMPs

Board 25

Board 25
Opponents vulnerable

♠ 8 7 6 3 2 ♥ J 3 ♦ 8 5 ♣ A Q 9 6

Partner opens one diamond in first seat, and I respond one spade. Partner bids one notrump.  This is too weak a hand to pass one notrump, and it's been decades since one could rebid a non-forcing two clubs.  I have little choice but to bid two spades.  I just hope partner isn't 1-4-4-4 or 1-4-5-3.  Two spades ends the auction, and West leads the seven of clubs:

	NORTH ♠ A 10 ♥ K 5 4 ♦ A J 4 3 2 ♣ 5 4 2
	SOUTH ♠ 8 7 6 3 2 ♥ J 3 ♦ 8 5 ♣ A Q 9 6

West	North	East	South
	1 ♦	Pass	1 ♠
Pass	1 NT	Pass	2 ♠
(All pass)

I have at least two spade losers, and one loser in each of the other suits. I can't afford any more. That means I need the heart ace onside and three-three spades. I also need to find something to do with my fourth club. I play the deuce of clubs from dummy, and East plays the king, and I win with the ace.

If clubs are four-two, I may need to set up a diamond trick for a club discard. It's tempting to play diamonds now. Unfortunately, I can't afford to. Say I duck a diamond and the defense continues clubs. I now play ace and a trump. If trumps are three-three (as they need to be), the opponents can separate their trump tricks via a club ruff. So I need to get trumps out of the way first.

I play a spade to the ace. West plays the four; East, the nine. I play dummy's ten of spades to East's queen, while West plays the five. East returns the ten of clubs, I win with the queen, and West follows with the three. So clubs are 4-2. It's possible West began with a doubleton and East's ten is a falsecard, but I doubt it.  I'm going to assume West began with J873.  If so, it's not going to be easy to avoid a second club loser. It appears I need the king-queen of diamonds to be onside. Let's say I play a third trump, and the opponents play ace and a heart. I ruff dummy's last heart to my hand. I'm down to:

	NORTH ♠ -- ♥ -- ♦ A J 4 3 ♣ 5
	SOUTH ♠ 8 ♥ -- ♦ 8 5 ♣ 9 6

If West has king-queen of diamonds and J8 of clubs, I can play a diamond and duck if he plays an honor. Do I have any other chances?  Maybe.  If I'm confident West is out of hearts in this end position, it doesn't hurt to cash the last trump. If West has king third of diamonds (without the queen), he may think he's being squeezed.  From his point of view, I might have:

♠ x ♥ -- ♦ Q x x ♣ x

If so, he can't afford to pitch a diamond. Nor can he afford the eight of clubs, since I could then endplay him. Maybe he'll decide to play his partner for the nine of clubs and throw the jack. If he does, I'll not only make the contract, I'll make an overtrick.

I play the third round of spades. West plays the jack, I pitch the deuce of diamonds from dummy, and East plays the deuce of hearts. I can no longer make this. The best I can hope for is down one. West cashes his last trump, I discard dummy's three of diamonds, and East discards the six of hearts. West now cashes the jack of clubs, solving my club problem. The ace of hearts is onside, so I'm down only one:

	NORTH ♠ A 10 ♥ K 5 4 ♦ A J 4 3 2 ♣ 5 4 2
WEST ♠ K J 5 4 ♥ A Q ♦ K 10 7 ♣ J 8 7 3		EAST ♠ Q 9 ♥ 10 9 8 7 6 2 ♦ Q 9 6 ♣ K 10
	SOUTH ♠ 8 7 6 3 2 ♥ J 3 ♦ 8 5 ♣ A Q 9 6

West apparently was worried about my having queen third of diamonds and a singleton club.  Rather than wait to discard the nine of clubs in the diagrammed position, he chose to cash it now.  This would be necessary if my singleton club were the nine.

The auction is the same at the other table. But declarer takes a different approach to the play. He wins the club lead and plays a spade to the ten. This seems wrong. You need three-three spades, so why block the suit? East wins and returns a club. Declarer wins and plays a diamond to the jack. We saw the danger of playing diamonds too early, and our fears proved correct.  East takes his queen and crosses to his partner's heart ace. West plays jack and a club, promoting an extra trump trick for the defense. Down two.

Me: -50
Jack: -100

Score on Board 25: +2 IMPs
Total: +76 IMPs

Board 24

Board 24
Neither vulnerable

♠ Q J 9 4 3 ♥ Q 8 7 6 ♦ Q 5 ♣ 10 6

RHO opens one notrump in third seat. John used to say that opening a strong notrump in front of me was like waving a red flag in front of a bull. But, red flag or not, I'm not charging without a singleton. I pass. LHO bids two clubs, RHO bids two hearts, and LHO bids three notrump.

The opponents have shown both my suits, so it's tempting to lead a minor. But partner could have doubled two clubs, and he could have opened with a weak two-diamond bid. So I doubt that either of his minors is good enough for a lead there to be productive.

Of my own suits, spades requires the least help from partner, so I decide to lead one. In general, you should lead your lowest card when leading one of your opponents' suits in notrump. There's no point in leading the top of a sequence. If partner has an honor, he's going to need to unblock. If he doesn't have an honor, you've probably led the wrong suit anyway, so what's the difference? Often even your fourth best is a card you can't afford to squander. Obviously, you can afford fourth best when your fourth best and lowest are equals as they are here. But, since partner can't expect you to lead fourth best, you might as well lead lowest even then for consistency.  Accordingly, I lead the three of spades.

	NORTH ♠ A 8 5 2 ♥ J 4 ♦ A 10 6 ♣ J 9 7 5
WEST ♠ Q J 9 4 3 ♥ Q 8 7 6 ♦ Q 5 ♣ 10 6

West	North	East	South
	Pass	Pass	1 NT
Pass	2 ♣	Pass	2 ♥
Pass	3 NT	(All pass)

Declarer plays the deuce from dummy, partner plays the ten, and declarer wins with the king. Partner has from six to eight high-card points. I need to hope he gains the lead before I do so he can continue spades (if he has one) while I still have the queen of hearts as a potential entry.

I get my wish. Declarer plays the deuce of clubs--six--jack--king.

This doesn't look promising. Declarer is perfectly happy to lose a club trick when he could have tried to play the suit for no losers. If he were in trouble, he probably wouldn't be playing this way. How many clubs did declarer start with? Might he have opened one notrump with a 2-4-2-5 pattern?

----

Surely not.  I don't know about his bidding style, but this is hardly the right way to play with ace-queen-fifth of clubs. He would be losing two tricks unnecessarily if the suit split 4-0. So declarer has three clubs tricks, two spade tricks, and at most three tricks in the red suits. (Partner must have a red king or the heart ace to come to six high-card points.) He's still a trick short.

Partner returns the six of spades--seven--nine (the card I'm known to hold, since partner played the ten at trick one)--five. If declarer was going to duck a spade, he might have done better to duck the first one. I continue with the queen of spades--ace--diamond deuce--heart deuce. Declarer plays a club to his ace and a club back to dummy's nine. Partner plays three-eight. I follow once, then pitch the six of hearts. Declarer leads the jack of hearts from dummy, partner covers with the king, declarer ducks (playing the ten), and I follow with the seven. Partner continues with the nine of hearts (Yes, the nine. I, too, thought declarer had that card.), declarer wins with the ace.

Since the queen and eight are equals and declarer "knows" I have the queen from partner's play of the king, I would normally play the queen, the card I'm known to hold. In this case, though, I want declarer to know I began with four hearts. I don't want to give him any reason not to play partner for the queen of diamonds. So I play the eight. Declarer plays a club to dummy. I pitch a spade, and partner pitches the three of diamonds. As expected, declarer cashes the diamond ace and plays a diamond to his jack. I claim for down two.

	NORTH ♠ A 8 5 2 ♥ J 4 ♦ A 10 6 ♣ J 9 7 5
WEST ♠ Q J 9 4 3 ♥ Q 8 7 6 ♦ Q 5 ♣ 10 6		EAST ♠ 10 6 ♥ K 9 3 ♦ 9 8 4 3 2 ♣ K 8 3
	SOUTH ♠ K 7 ♥ A 10 5 2 ♦ K J 7 ♣ A Q 4 2

I know what you're going to say. I was supposed to falsecard with the queen of hearts anyway, since a suspicious declarer might ask himself why I didn't and decide to play me for the diamond queen. Actually, if any declarer demonstrated that much confidence that I'm incapable of making a careless play, I would be so flattered that I wouldn't mind letting him make this contract.

At the other table, the auction was the same. My hand led the ten of clubs, picking up four club tricks for declarer. On the fourth round of clubs, both defenders pitched a diamond. Declarer led a diamond to the king, dropping the queen, and a diamond back to dummy, on which West pitched a heart. When East failed to cover the jack of hearts, declarer had three heart tricks. He wound up making six. A five-trick difference on the choice of opening leads! Well, maybe not entirely on the choice of opening leads. Anyway, all the overtricks are irrelevant. Minus 400 would have been the same 11 imps.

Me: +100
Jack -490

Score on Board 24: +11 IMPs
Total: + 74 IMPs